Application of the boundary integral equation method to very small wavelength-to-period diffraction problems

Abstract

Diffraction from 1D multilayer gratings having arbitrary border profiles including edges is considered for small wavelength-to-period (λ/d) ratios, the most difficult case for any rigorous numerical method. The boundary integral equation theory is so flexible that we can indicate a few areas where it can be modified. In this work, special attention is paid to the main aspects of the Modified Boundary Integral Equation Method for λ/d ≪ 1 as well as to a more general treatment of the energy conservation law applicable to multilayer absorption gratings. Three types of small λ/d problems are known from optical applications: (a) shallow gratings working in the X-ray and extreme ultraviolet ranges, both at near-normal and grazing angles, (b) deep echelle gratings with a steep working facet illuminated along its normal by light of any wavelength, and (c) rough mirrors and gratings in which rough boundaries can be represented by a large-d grating, and which contain a number of random asperities illuminated at any angle and wavelength. Numerical examples of diverse in-plane diffraction problems are presented.

Acknowledgements

The author feels indebted to Sergey Sadov (Newfoundland Memorial University, Canada) for the information provided and fruitful collaboration. Helpful discussions and testing with Bernd Kleemann (Carl Zeiss AG, Germany), Andreas Rathsfeld (WIAS, Germany), and Gunther Schmidt (WIAS, Germany) are greatly appreciated.